Absolute Value Calculator
With a — inside the bars, x coefficient 3, b — inside the bars, constant -7, c — what it equals 5, absolute value comes to 4.0000 — x, taking the positive case. It is reached in 6 steps, the last of which is (5 - -7) / 3, and each one is printed on the page with its numbers filled in. The formula is the one published by NIST Digital Library of Mathematical Functions, not an approximation fitted to it.
The absolute value of a number, and the two solutions an equation of the form |ax + b| = c always has — which is the part a bare magnitude does not answer.
Formula and sources checked · How we check
a — inside the bars, x coefficient 3, b — inside the bars, constant -7, c — what it equals 5
4.0000
x, taking the positive case for the example below. Editing a field recomputes the calculator below; this figure holds the answer the page was loaded with.
It is written into the HTML rather than drawn by a script, so a search engine reading this page without running JavaScript still finds an answer.
- Solving ax + b = c
(5 - -7) / 34- Solving ax + b = −c
(-5 - -7) / 30.667- Midpoint of the two, −b ÷ a
--7 / 32.333- Distance from the midpoint to each
abs(5 / 3)1.667- The expression at x = 0, |b|
abs(-7)7- Number of real solutions
2
Worked example
|3x − 7| = 5 splits into 3x − 7 = 5 and 3x − 7 = −5, giving x = 4 and x = 2/3. Both are real solutions and dropping the second is the standard mistake: the two sit either side of x = 7/3, each 5/3 away.
How to work it out yourself
- 1.Split into two equations, always. |u| = c means u = c or u = −c, so an absolute value equation with a positive right-hand side has two solutions and reporting one is reporting half the answer.
- 2.Check the sign of c before solving. A negative right-hand side has no solution at all, because an absolute value is a distance and a distance is never negative. c = 0 is the one case with a single solution.
- 3.Read the geometry: the solutions sit either side of x = −b/a, each |c/a| away. That is the same statement as "|ax + b| is a distance", and it is how to sanity-check an answer.
The formula
- Solving ax + b = c
(5 - -7) / 3 - Solving ax + b = −c
(-5 - -7) / 3 - Midpoint of the two, −b ÷ a
--7 / 3 - Distance from the midpoint to each
abs(5 / 3) - The expression at x = 0, |b|
abs(-7) - Number of real solutions
Source: NIST Digital Library of Mathematical Functions — absolute value, OpenStax College Algebra — linear equations with absolute value
Questions people actually ask
- What is the absolute value of a number?
- Its distance from zero, without direction: |−7| and |7| are both 7. That definition is why an absolute value is never negative and why an equation involving one usually has two solutions.
- Why does |3x − 7| = 5 have two answers?
- Because two different quantities have an absolute value of 5: 5 and −5. Setting 3x − 7 to each gives x = 4 and x = 2/3, and both satisfy the original equation. Any absolute value equation with a positive right-hand side works the same way.
- When does an absolute value equation have no solution?
- When it equals a negative number. |x + 2| = −3 has none, because the left side is a distance and cannot be negative. There is no algebra to do — the contradiction is in the statement.
Related
- Quadratic Formula CalculatorBoth roots of ax² + bx + c = 0 from the quadratic formula, with the discriminant that says in advance how many real roots there are.
- Completing the Square CalculatorTurns ax² + bx + c into a(x − h)² + k, giving the vertex, the axis of symmetry and the roots that fall straight out of the completed form.
- Percent Error CalculatorPercent error against a known value, with percent difference beside it — they are different questions and get confused constantly.
- Distance Formula CalculatorDistance between two points in the plane or in space, with the midpoint and the slope, and every step of the square root shown.
- Difference of Two SquaresFactors a² − b² into (a + b)(a − b), multiplied back out as a check — and the mental shortcut it gives for products either side of a round number.
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